Initial Value Methods for Axial Dispersion Problems

The initial value method is a useful method for two-point boundary value problems, including reactor problems with axial dispersion.

Consider first the case of a reaction that is second order in concentration.

The initial value method translates this second-order equation into two first-order equations which can be integrated using ODE solvers. In this case we define the new variables

and the equations become

The boundary conditions are

In approach one, we guess values at the inlet, integrate to the end, and check if the last condition is satisfied. If not, we change the guess and repeat the operation. Here the boundary conditons are taken as

In order to use a minimum number of guesses, we use the Newton Raphson method (link) for the iterations. We thus want so choose alpha so that the following function is zero.

In order to apply the Newton Raphson method we need the derivative of psi with respect to alpha, and that requires that we know the derivative of y₂(1,alpha) with respect to alpha. We obtain this value by differentiating all the equations with respect to alpha, and defining new variables.

The additional equations are then

and the additional boundary conditions are

This enables us to calculate the derivative

The iterations proceed using the formula

The code is available. When the code was run, however, it was found that the iteration is unstable. Solutions could be obtained for Da = 1 and Pe = 1, 2, 3, 4.5, and 4.6, but not 4.7 or higher values. The progression of solutions is shown in the figure.

Fortunately, the iterations do converge if the integration is done in the reverse direction. Thus, in the second approach, we integrate from the outlet to the inlet. Thus, make the variable change: z = ­ x, and the equations are

When they are converted to two first-order equations, the problem is

We want to choose the value of concentration at the outlet

and check that the boundary condition at the inlet is satisfied.

Thus, we introduce new variables

and derive their equations, as before.

The additional boundary conditions are

Now the derivative of psi with respect to alpha is

and the iterations proceed following

The

The code is available.

Now the iterations do converge, and the solution is shown in the two figures.

This method will not always work (ref), but is a quick, efficient method to use when it does work.

Also, see the other solution methods: orthogonal collocation, finite difference, orthogonal collocation on finite elements.

Take Home Message: The initial value method is easy to use since it employs ODE solvers which are readily available. One must integrate from the exit to the entrance of the reactor.